Global pluripotential theory on hybrid spaces

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Let (X,L) be a polarized scheme over a Banach ring A. We define and study a class PSH(X,L) of plurisubharmonic metrics on the Berkovich analytification X an . We focus mainly on the case where A is a hybrid ring of power series, so that X an is the hybrid space associated to a degeneration of complex manifolds X. We then prove that any plurisubharmonic metric on (X,L) with logarithmic growth at zero admits a canonical plurisubharmonic extension to the hybrid space X hyb . We also discuss the continuity of the family of Monge-Ampère measures associated to a continuous plurisubharmonic hybrid metric.